Little orthogonality theorem

The little orthogonality theorem consists of two relations that are reduced forms of the mathematical expression for the great orthogonality theorem.

1st little orthogonality relation

The 1st relation is derived from eq21 by letting  and , resulting in

Since , we have  and


Show that .



Since the traces of a matrices of the same class are the same, eq22 is equivalent to


    1. is the number of classes.
    2. is the number of elements in the -th class.
    3. , called the character of a class, is the trace of a matrix belonging to the -th class of the -th irreducible representation.

Eq23 is known as the 1st little orthogonality relation.


Determine whether the representations  and  of the point group are reducible or irreducible.


Since the 1st little orthogonality relation is derived from the great orthogonality theorem, which pertains only to irreducible representations, characters of a representation that do not satisfy the relation belong to a reducible representation. Using eq23,

Therefore,  is an irreducible representation of the  point group, while  is a reducible representation of the  point group.


Let’s rewrite eq23 as

Eq24 has the form of the inner product of two vectors  and  in a -dimensional vector space, with components  and respectively. The components are functions of  and the two vectors are orthogonal to each other when . A -dimensional vector space is spanned by  orthogonal vectors, each with  components. Since the number of vectors and the number of components of each vector are denoted by  and  respectively, the number of irreducible representations of a group is equal to the number of classes of that group. We say that the irreducible representations of a group form a complete set of basis vectors in the -dimensional vector space, with the components of each basis vector being .

2nd little orthogonality relation

Consider the matrices  and  with entries  and respectively.

Both are square matrices because the number of irreducible representations of a group is equal to the number of classes of that group. Comparing with eq24, the entries of the matrix product are given by:

Since  and  are square matrices and , then , i.e.

Eq24b is the 2nd little orthogonality relation.


Show that if  and  are square matrices and , then .


Using the determinant identities of and , we have , which implies that and are not zero and are therefore non-singular. So,


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